1. Spanning Trees Longin Jan Latecki Temple University based on slides by
David Matuszek, UPenn,
Rose Hoberman, CMU,
Bing Liu, U. of Illinois,
Boting Yang, U. of Regina

2. Spanning trees Suppose you have a connected undirected graph
Connected: every node is reachable from every other node
Undirected: edges do not have an associated direction
...then a spanning tree of the graph is a connected subgraph in which there are no cycles
A connected, undirected graph
Four of the spanning trees of the graph

3. Theorem
A simple graph is connected iff it has a spanning tree.

4. Finding a spanning tree
To find a spanning tree of a graph,
pick an initial node and call it part of the spanning tree
do a search from the initial node:
each time you find a node that is not in the spanning tree, add to the spanning tree both the new node and the edge you followed to get to it
An undirected graph
One possible result of a BFS starting from top
One possible result of a DFS starting from top

5. Graph Traversal Algorithm
To traverse a tree, we use tree traversal algorithms like pre-order, in-order, and post-order to visit all the nodes in a tree
Similarly, graph traversal algorithm tries to visit all the nodes it can reach.
If a graph is disconnected, a graph traversal that begins at a node v will visit only a subset of nodes, that is, the connected component containing v.

6. Two basic traversal algorithms
Two basic graph traversal algorithms:
Depth-first-search (DFS)
After visit node v, DFS strategy proceeds along a path from v as deeply into the graph as possible before backing up
Breadth-first-search (BFS)
After visit node v, BFS strategy visits every node adjacent to v before visiting any other nodes

7. Depth-first search (DFS)
DFS strategy looks similar to pre-order. From a given node v, it first visits itself. Then, recursively visit its unvisited neighbors one by one.
DFS can be defined recursively as follows.
procedure DSF(G: connected graph with vertices v1, v2, …, vn)
T:=tree consisting only of the vertex v1
visit(v1)
mark v1 as visited;
procedure visit(v: vertex of G)
for each node w adjacent to v not yet in T
begin
add vertex w and edge {v, w} to T
visit(w);
mark w as visited;
end

8. x x x x x DFS example v3 v2 v2 v3 v1 v1 v4 v5 v4 G v5 Start from v3 1

9. Breadth-first search (BFS)
BFS strategy looks similar to level-order. From a given node v, it first visits itself. Then, it visits every node adjacent to v before visiting any other nodes.
1. Visit v
2. Visit all v’s neighbors
3. Visit all v’s neighbors’ neighbors …
Similar to level-order, BFS is based on a queue.

10. Algorithm for BFS
procedure BFS(G: connected graph with vertices v1, v2, …, vn)
T:= tree consisting only of vertex v1
L:= empty list
put v1 in the list L of unprocessed vertices;
while L is not empty
begin
remove the first vertex v from L
for each neighbor w of v
if w is not in L and not in T then
add w to the end of the list L
add w and edge {v, w} to T
end

11. x x x x x BFS example v5 v2 v3 v1 v3 v4 v4 v2 v5 G v1 Start from v5 1
Visit
Queue (front to back) v5
empty v3 v4
v3, v4 v2
v4, v2 v1 1 v5 v1 v4 v3 v5 v2 G v3 2 v4 3 x x x v2 4 x x v1 5

12. Backtracking Applications
Use backtracking to find a subset of {31, 27, 15, 11, 7, 5} with sum equal to 39.